Geodesic
Triple covers and a non-simply connected surface spanning an elongated tetrahedron and beating the cone
Giovanni Bellettini1 ; Maurizio Paolini2 ; Franco Pasquarelli2
1Università di Siena, Italy and International Center for Theoretical Physics, Trieste, Italy
2Università Cattolica del Sacro Cuore, Brescia, Italy
Interfaces and free boundaries, Tome 20 (2018) no. 3, pp. 407-436

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DOI

By using a suitable triple cover we show how to possibly model the construction of a minimal surface with positive genus spanning all six edges of a tetrahedron, working in the space of BV functions and interpreting the film as the boundary of a Caccioppoli set in the covering space. After a question raised by R. Hardt in the late 1980’s, it seems common opinion that an area-minimizing surface of this sort does not exist for a regular tetrahedron, although a proof of this fact is still missing. In this paper we show that there exists a surface of positive genus spanning the boundary of an elongated tetrahedron and having area strictly less than the area of the conic surface.
DOI : 10.4171/ifb/407
Classification : 49-XX, 57-XX
Mots-clés : Plateau problem, soap films, covering spaces

Giovanni Bellettini  1   ; Maurizio Paolini  2   ; Franco Pasquarelli  2

1 Università di Siena, Italy and International Center for Theoretical Physics, Trieste, Italy
2 Università Cattolica del Sacro Cuore, Brescia, Italy 
Giovanni Bellettini; Maurizio Paolini; Franco Pasquarelli. Triple covers and a non-simply connected surface spanning an elongated tetrahedron and beating the cone. Interfaces and free boundaries, Tome 20 (2018) no. 3, pp. 407-436. doi: 10.4171/ifb/407
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     journal = {Interfaces and free boundaries},
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     year = {2018},
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     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/407/}
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